# 3.1 Signed numbers

 Page 1 / 1
This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr. The basic operations with real numbers are presented in this chapter. The concept of absolute value is discussed both geometrically and symbolically. The geometric presentation offers a visual understanding of the meaning of |x|. The symbolic presentation includes a literal explanation of how to use the definition. Negative exponents are developed, using reciprocals and the rules of exponents the student has already learned. Scientific notation is also included, using unique and real-life examples.Objectives of this module: be familiar with positive and negative numbers and with the concept of opposites.

## Overview

• Positive and Negative Numbers
• Opposites

## Positive and negative numbers

When we studied the number line in Section [link] we noted that

Each point on the number line corresponds to a real number, and each real number is located at a unique point on the number line. ## Positive and negative numbers

Each real number has a sign inherently associated with it. A real number is said to be a positive number if it is located to the right of 0 on the number line. It is a negative number if it is located to the left of 0 on the number line.

## The notation of signed numbers

A number is denoted as positive if it is directly preceded by a $"+"$ sign or no sign at all.
A number is denoted as negative if it is directly preceded by a $"-"$ sign.

The $"+"$ and $"-"$ signs now have two meanings:

$+$ can denote the operation of addition or a positive number.
$-$ can denote the operation of subtraction or a negative number.

## Read the $"-"$ Sign as "negative"

To avoid any confusion between "sign" and "operation," it is preferable to read the sign of a number as "positive" or "negative."

## Sample set a

$-8$ should be read as "negative eight" rather than "minus eight."

$4+\left(-2\right)$ should be read as "four plus negative two" rather than "four plus minus two."

$-6+\left(-3\right)$ should be read as "negative six plus negative three" rather than "minus six plusminus three."

$-15-\left(-6\right)$ should be read as "negative fifteen minus negative six" rather than "minus fifteenminus minus six."

$-5+7$ should be read as "negative five plus seven" rather than "minus five plus seven."

$0-2$ should be read as "zero minus two."

## Practice set a

Write each expression in words.

$4+10$

four plus ten

$7+\left(-4\right)$

seven plus negative four

$-9+2$

negative nine plus two

$-16-\left(+8\right)$

negative sixteen minus positive eight

$-1-\left(-9\right)$

negative one minus negative nine

$0+\left(-7\right)$

zero plus negative seven

## Opposites

On the number line, each real number has an image on the opposite side of 0. For this reason we say that each real number has an opposite. Opposites are the same distance from zero but have opposite signs.

The opposite of a real number is denoted by placing a negative sign directly in front of the number. Thus, if $a$ is any real number, then $-a$ is its opposite. Notice that the letter $a$ is a variable. Thus, $"a"$ need not be positive, and $"-a"$ need not be negative.

If $a$ is a real number, $-a$ is opposite $a$ on the number line and $a$ is opposite $-a$ on the number line. $-\left(-a\right)$ is opposite $-a$ on the number line. This implies that $-\left(-a\right)=a$ .

This property of opposites suggests the double-negative property for real numbers.

## The double-negative property

If $a$ is a real number, then
$-\left(-a\right)=a$

## Sample set b

If $a=3$ , then $-a=-3$ and $-\left(-a\right)=-\left(-3\right)=3$ . If $a=-4$ , then $-a=-\left(-4\right)=4$ and $-\left(-a\right)=a=-4$ . ## Practice set b

Find the opposite of each real number.

8

$-8$

17

$-17$

$-6$

6

$-15$

15

$-\left(-1\right)$

$-1$ , since $-\left(-1\right)=1$

$-\left[-\left(-7\right)\right]$

7

Suppose that $a$ is a positive number. What type of number is $-a$ ?

If $a$ is positive, $-a$ is negative.

Suppose that $a$ is a negative number. What type of number is $-a$ ?

If $a$ is negative, $-a$ is positive.

Suppose we do not know the sign of the number $m$ . Can we say that $-m$ is positive, negative, or that we do notknow ?

We must say that we do not know.

## Exercises

A number is denoted as positive if it is directly preceded by ____________________ .

a plus sign or no sign at all

A number is denoted as negative if it is directly preceded by ____________________ .

For the following problems, how should the real numbers be read ? (Write in words.)

$-5$

a negative five

$-3$

12

twelve

10

$-\left(-4\right)$

negative negative four

$-\left(-1\right)$

For the following problems, write the expressions in words.

$5+7$

five plus seven

$2+6$

$11+\left(-2\right)$

eleven plus negative two

$1+\left(-5\right)$

$6-\left(-8\right)$

six minus negative eight

$0-\left(-15\right)$

Rewrite the following problems in a simpler form.

$-\left(-8\right)$

$-\left(-8\right)=8$

$-\left(-5\right)$

$-\left(-2\right)$

2

$-\left(-9\right)$

$-\left(-1\right)$

1

$-\left(-4\right)$

$-\left[-\left(-3\right)\right]$

$-3$

$-\left[-\left(-10\right)\right]$

$-\left[-\left(-6\right)\right]$

$-6$

$-\left[-\left(-15\right)\right]$

$-\left\{-\left[-\left(-26\right)\right]\right\}$

26

$-\left\{-\left[-\left(-11\right)\right]\right\}$

$-\left\{-\left[-\left(-31\right)\right]\right\}$

31

$-\left\{-\left[-\left(-14\right)\right]\right\}$

$-\left[-\left(12\right)\right]$

12

$-\left[-\left(2\right)\right]$

$-\left[-\left(17\right)\right]$

17

$-\left[-\left(42\right)\right]$

$5-\left(-2\right)$

$5-\left(-2\right)=5+2=7$

$6-\left(-14\right)$

$10-\left(-6\right)$

16

$18-\left(-12\right)$

$31-\left(-1\right)$

32

$54-\left(-18\right)$

$6-\left(-3\right)-\left(-4\right)$

13

$2-\left(-1\right)-\left(-8\right)$

$15-\left(-6\right)-\left(-5\right)$

26

$24-\left(-8\right)-\left(-13\right)$

## Exercises for review

( [link] ) There is only one real number for which ${\left(5a\right)}^{2}=5{a}^{2}$ . What is the number?

0

( [link] ) Simplify $\left(3xy\right)\left(2{x}^{2}{y}^{3}\right)\left(4{x}^{2}{y}^{4}\right)$ .

( [link] ) Simplify ${x}^{n+3}\cdot {x}^{5}$ .

${x}^{n+8}$

( [link] ) Simplify ${\left({a}^{3}{b}^{2}{c}^{4}\right)}^{4}$ .

( [link] ) Simplify ${\left(\frac{4{a}^{2}b}{3x{y}^{3}}\right)}^{2}$ .

$\frac{16{a}^{4}{b}^{2}}{9{x}^{2}{y}^{6}}$

where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
Please keep in mind that it's not allowed to promote any social groups (whatsapp, facebook, etc...), exchange phone numbers, email addresses or ask for personal information on QuizOver's platform. By By OpenStax By OpenStax By Robert Murphy By Inderjeet Brar By Caitlyn Gobble By Michael Nelson By Stephen Voron By OpenStax By Jonathan Long By Rachel Woolard